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Mixed finite element methods for a heat-assisted model of sedimentation

Julio Careaga, Cristian Inzunza

arXiv:2610.28903v1math.NAcs.NA

TL;DR

Temperature affects material properties and flow in sedimentation, motivating a coupled concentration, heat, and flow model. The paper analyzes mixed formulations and finite element approximations; it establishes continuous-system uniqueness under stated conditions and reports first-order convergence for most variables.

  • Problem

    Temperature changes material properties and fluid flow in sedimentation, so the model must include a temperature equation alongside concentration and flow.

    Source: §1.1 · pp. 1–2

  • Method

    The paper formulates the quasi-stationary coupled system using mixed saddle-point problems and finite element spaces, then analyzes solvability and approximation.

    Source: §1.1 · p. 3, §8 · pp. 34–35

  • Results

    The fully coupled continuous system has a unique solution under the stated conditions, and the accuracy test reports first-order convergence for all variables except the trace variable.

    Source: §4.5 · p. 21, §7.1 · p. 30

  • Takeaways & Limitations

    The numerical examples demonstrate the mixed finite element scheme and report convergence behavior consistent with the theoretical estimates for the lowest-order method.

    Source: §1.1 · p. 3, §7.1 · p. 30

    The continuous uniqueness analysis requires an additional regularity assumption that some solution components belong to a slightly smoother Sobolev space.

    Source: §8 · p. 35

Abstract

from arXiv · show

A model for the consolidation-sedimentation process incorporating thermal effects is analysed. The model is motivated by novel applications to heat-assisted sedimentation of mineral suspensions. A system of convection-diffusion equations for the concentration and temperature are coupled to flow equations for the volume-average velocity and pressure. The model is supplemented with Dirichlet and mixed boundary conditions. A quasi-stationary approach of the system is analysed by means of a mixed variational formulation, where additional variables such as the pseudostress tensor and variables for the gradient of the temperature and concentration are introduced. The existence and uniqueness of the system is shown making use of the Babu{\v s}ka-Brezzi theory and Banach's fixed-point theorem, under hypotheses of smallness of the data and additional regularity on the solutions. A discrete formulation based on finite element spaces is proposed and analysed, and the existence of solutions is achieved thanks to Brouwer's fixed-point theorem. Additionally, C\'ea and a priori estimates are derived for the mixed formulation. Finally, numerical examples are provided to demonstrate the performance of the model and to confirm the theoretical rates of convergence.

1 Introduction

The paper studies a quasi-stationary, thermally coupled model of multidimensional sedimentation-consolidation, where temperature affects fluid properties and flow. It develops mixed formulations and finite element analysis for the coupled system.

1.1 Scope

  • Temperature changes material properties and therefore affect fluid flow and sedimentation, motivating an additional temperature equation.The model couples concentration and temperature convection-diffusion equations to flow equations for volume-average velocity and pressure.Source: §1.1 · pp. 1–2
  • The model combines hindered particle motion and solid-phase compression with viscosity that varies with concentration and temperature.Concentration also influences suspension heat capacity and heat transfer; the flow source includes a temperature-dependent contribution.Source: §1.1 · p. 2
  • The authors present a first analysis of this coupled multiphysics problem and aim to establish mixed-formulation and finite element results.The stated aims include existence and a priori estimates, numerical investigation of convergence rates, and simulations.Source: §1.1 · p. 3

1.2 Related works

  • Prior multidimensional sedimentation models couple concentration transport to flow, while temperature-inclusive models treat related heat-assisted settings.The introduction positions this work among studies of sedimentation-consolidation, coupled flow and convection-diffusion, and temperature-varying sedimentation.Source: §1.2 · pp. 3–4

1.3 Outline of the paper

  • The paper proceeds from governing equations and mixed weak formulations to solvability, discrete existence, a priori estimates, and numerical examples.The outline identifies fixed-point analysis for the continuous system and numerical examples intended to show performance and theoretical convergence rates.Source: §1.3 · p. 4

2 Model formulation

The model is a quasi-stationary coupled PDE system for concentration, temperature, velocity, and pressure, with functions and boundary data subject to stated assumptions. The formulation specifies the functional-space notation used for its analysis.

2.1 Further notation

  • The notation uses L^k(Ω) for Lebesgue spaces and W^{k,s}(Ω) for Sobolev spaces, with corresponding norms and seminorms.The domain is a bounded polyhedral subset of R^d, where d is 2 or 3.Source: §2.1 · p. 4
  • The text defines vector and tensor spaces and distinguishes divergence on vector fields from its row-wise tensor extension.It also specifies trace notation and norms for the listed spaces.Source: §2.1 · p. 5
  • Boundary duality pairings use trace spaces and extensions, while the integration-by-parts identities apply to stated H(div) spaces.The passage also defines tensor transpose, deviatoric part, and symmetric, skew-symmetric, and zero-trace tensor notation.Source: §2.1 · p. 5
  • The notation includes a second integration-by-parts formula for tensors and vectors, with “:” denoting the matrix inner product.Duality pairings on relatively open boundary subsets follow the stated trace-space convention.Source: §2.1 · p. 5

2.2 Governing equations

  • The unknowns are solid volume fraction ϕ, volume-average velocity u, excess pore pressure p, and temperature θ.For the monodispersed model, the fluid volume fraction is 1 − ϕ.Source: §2.2 · p. 6
  • The model uses Kynch flux for hindered settling, D1 for compression, D2 for thermal diffusivity, and concentration- and temperature-dependent viscosity.Permeability depends on position and concentration, and the flow source depends on both concentration and temperature.Source: §2.2 · p. 6
  • The governing system couples concentration and temperature convection-diffusion equations with momentum balance and velocity continuity.The equations are interpreted as quasi-stationary, with selected terms related to backward Euler discretizations of time derivatives.Source: §2.2 · p. 6
  • The functions and parameters are constrained by bounds, positivity, uniform positive definiteness, and Lipschitz-continuity assumptions.In particular, Assumption 1 bounds the flux and diffusion functions and specifies positive lower bounds for D1 and D2.Source: §2.2 · pp. 6–7

3 Continuous saddle-point problem formulations

The continuous formulation rewrites concentration, heat, and flow equations as mixed saddle-point problems by introducing fluxes, gradients, and stress-related variables. These formulations specify the spaces and boundary terms used for the coupled analysis.

3.1 Concentration equation

  • The concentration formulation introduces flux z and gradient r so that concentration ϕ can be sought in a weaker space than H1(Ω).The flux combines convection, hindered-settling, and diffusion contributions, while r represents ∇ϕ.Source: §3.1 · p. 7
  • The concentration spaces and estimates require exponents r and ρ above 2 in two dimensions and within [3, 6] in three dimensions.The convective product requires ϕu ∈ L2(Ω), and the stated exponent ranges ensure the needed embeddings.Source: §3.1 · p. 7
  • The concentration weak formulation is a nonlinear perturbed saddle-point problem on product spaces for flux-gradient variables and concentration-boundary variables.Its construction tests the gradient relation with H(div) functions and incorporates the boundary unknown ξ := −γ0(ϕ).Source: §3.1 · pp. 7–8

3.2 Heat equation

  • The heat formulation introduces heat flux eσ and temperature gradient t, with t = ∇θ and eσ = −θu + D2(ϕ)t.The resulting heat equation imposes Dirichlet temperature data on ΓD and zero normal heat flux on ΓN.Source: §3.2 · p. 9
  • The heat equation is posed as a perturbed saddle-point problem for temperature-gradient and scalar variables, given velocity u and concentration ϕ.Testing and integration by parts yield the weak formulation, including the boundary pairing with prescribed temperature on ΓD.Source: §3.2 · p. 9

3.3 Flow equations

  • The flow formulation introduces strain rate ζ, vorticity γ, and viscous stress σ, allowing pressure to be eliminated.The approach follows the mixed formulation cited as, and the symmetry of σ is imposed weakly.Source: §3.3 · p. 10
  • The flow equations are recast as a linear perturbed saddle-point problem on product spaces for stress-related variables and velocity-vorticity variables.The formulation is given for prescribed concentration ϕ and temperature θ.Source: §3.3 · p. 10

4 Continuous solvability analysis

The section establishes well-posedness for the linearized subproblems and uses their solution operators to analyze the fully coupled system. Under regularity and small-data conditions, it proves existence and uniqueness when the coupled fixed-point operator’s total Lipschitz constant is less than one.

4.1 Fixed-point operator

  • The three saddle-point solution operators are combined into a fixed-point operator T, and solving the coupled problem is equivalent to finding its fixed point.T acts on pairs of concentration and temperature functions in (Lρ(Ω))².Source: §4.1 · pp. 11–12

4.2 Well-posedness concentration equation

  • The concentration subproblem has a unique solution, defining S1, when the exponent condition holds and the velocity norm is sufficiently small.The convective term is treated as a sufficiently small perturbation.Source: §4.2 · pp. 13, 15

4.3 Well-posedness heat equation

  • The heat subproblem has a unique solution, defining S2, when the velocity norm is bounded by αA/2.The proof uses global inf-sup conditions for the perturbed bilinear form.Source: §4.3 · p. 16

4.4 Well-posedness flow equations

  • For admissible conjugate exponent pairs, the flow subproblem has a unique solution, defining S, for each concentration and temperature pair.The stability constant is independent of the concentration and temperature inputs.Source: §4.4 · p. 18

4.5 Solvability fully coupled system

  • Sufficiently small data make the coupled fixed-point operator a self-mapping on an ε-ball; its Lipschitz continuity is then studied under additional regularity assumptions.The operator is built from the solution operators for the concentration, heat, and flow subproblems.Source: §4.5 · pp. 18–19
  • Under hypothesis (H), the solution operators S1, S2, and S are Lipschitz continuous for their stated input domains.Lemmas 4.11–4.13 provide data- and input-independent Lipschitz constants.Source: §4.5 · p. 20
  • Under hypothesis (H) and the data smallness conditions, the operator T is Lipschitz continuous on Wε.Lemma 4.14 requires conjugate exponent pairs satisfying (3.4) and provides a constant LT.Source: §4.5 · p. 20
  • Theorem 4.15 establishes a unique fully coupled solution with concentration-temperature pair in Wε when the data satisfy (4.25) and LT < 1.The proof uses Banach’s fixed-point theorem for T on Wε; the remaining components follow from well-posedness of the decoupled problems.Source: §4.5 · p. 21

5 Discrete formulation and solvability analysis

The discrete scheme couples finite-element approximations of the concentration, heat, and flow subproblems; its analysis is restricted to two dimensions. Under the stated conditions, the decoupled problems are well posed and Brouwer’s theorem gives existence for the fully coupled discrete problem.

5.1 Discrete formulation

  • The discrete analysis is restricted to two dimensions, although the continuous analysis applies in both two and three dimensions.The finite-element spaces are built over regular triangular meshes.Source: §5.1 · p. 21
  • The scheme uses Raviart–Thomas spaces and AFW elements for flow, with a coarsened macro-edge boundary partition in the scalar spaces.The boundary edges are paired along straight boundary segments to form the coarsened partition.Source: §5.1 · p. 22

5.2 Solvability of the decoupled discrete problems

  • Discrete inf-sup conditions and positive semidefiniteness establish stability for the concentration saddle-point formulation, with constants independent of mesh size.The argument combines Raviart–Thomas divergence stability with the macro-edge construction of the boundary space.Source: §5.2 · p. 24
  • The concentration, heat, and flow discrete subproblems each have unique solutions under their stated hypotheses, defining the discrete solution operators.The concentration and heat results impose a bound on the velocity input; the flow result holds for each pair of scalar inputs.Source: §5.2 · pp. 24–25

5.3 Solvability of the fully coupled discrete problem

  • Unlike the continuous analysis, the discrete solvability argument requires no additional solution regularity, but finite-dimensional norm bounds are generally not uniform in mesh size.For each fixed mesh, finite-dimensionality suffices to establish continuity of the discrete solution operators.Source: §5.3 · p. 25
  • The discrete fixed-point operator maps a suitable ball into itself and is continuous there, yielding existence but not a uniqueness claim for the coupled discrete solution.Finite-dimensionality supplies continuity, but the associated constants need not be uniform in mesh size.Source: §5.3 · pp. 25–26
  • The fully coupled discrete problem has at least one solution when the stated exponent and data conditions hold, by applying Brouwer’s theorem to its discrete fixed-point operator.The operator maps a suitable closed ball into itself and is continuous there; the remaining discrete components are uniquely determined for each fixed point.Source: §5.3 · pp. 26–27

6 A priori error estimates

The section establishes a Cea-type bound for the total error and derives convergence rates under additional regularity of the continuous solution. The interior-variable rate is limited by the polynomial degree and the solution regularity.

Cea-type estimate

  • Theorem 6.1 gives a Cea-type bound for the total error under its stated hypotheses and effective discrete stability conditions.Its proof combines discrete inf-sup estimates and bounded bilinear forms with difference estimates for the coupled problems.Source: §6 · p. 27

Rate of convergence

  • Interior variables converge at order h^k when the stated regularity holds, with 1 ≤ k ≤ ℓ + 1.The proof uses Raviart–Thomas interpolation and local polynomial projectors; the boundary variable ξ requires a separate approximation argument.Source: §6 · p. 28
  • The boundary approximation uses continuous piecewise linear functions on the coarsened macro-edge partition and assumes ξ ∈ H^(1/2+δ)(Γ), with δ ∈ (0, 1].The passage identifies a corresponding boundary approximation property but does not state its rate in the supplied text.Source: §6 · p. 28

7 Numerical examples

The numerical examples test convergence with manufactured solutions and simulate batch and continuous sedimentation scenarios using the proposed scheme. Results include first-order convergence for most variables and time-dependent concentration, temperature, and velocity behavior.

7.1 Example 1: Accuracy test

  • All variables and total error converge first-order except ξh, whose H1/2(Γ) rate is approximately second-order; Picard tolerance was reached in five iterations.The tables report errors and experimental rates for concentration, heat, and flow unknowns over mesh sizes.Source: Table 7.1 · p. 29, §7.1 · p. 30, Table 7.2 · p. 30

7.2 Example 2: Batch sedimentation

  • By 5 s, solids are almost entirely separated; temperature approaches equilibrium, thermal influence on velocity is stronger early, and total mass is preserved at each time step.The batch Kynch test uses homogeneous initial conditions and cold and hot boundaries.Source: §7.2 · p. 31, Figure 7.3 · p. 32
  • In the Diehl test, solids initially in the vessel’s upper 30% move right, slide through the boundary, and settle at the bottom; heat reaches equilibrium by 3 s.The velocity field is initially circular, with greater magnitude along the heated right wall later.Source: §7.2 · p. 31, Figure 7.4 · p. 33

7.3 Example 3: Continuous sedimentation

  • In continuous sedimentation, concentration remains positive and bounded as solids enter at the top and leave at the right; temperature peaks at the outlet, with thermal influence visible near the outlet and inclined wall.The flow is primarily determined by the Dirichlet conditions at the inlet and outlet.Source: §7.3 · p. 34, Figure 7.5 · p. 34

8 Conclusions

The paper introduces a pseudo-steady heat-assisted sedimentation model and analyzes continuous and discrete mixed formulations. The analysis requires additional solution regularity, and several extensions remain future work.

8 Conclusions

  • The continuous analysis splits the coupled system into three linearized saddle-point problems, then uses Banach’s fixed-point theorem to establish existence and uniqueness.The saddle-point problems are analyzed with Babuška-Brezzi theory in Banach spaces.Source: §8 · p. 34
  • The contraction argument and coefficient-function couplings require additional regularity, with some solution components in a slightly smoother Sobolev space.The paper notes that similar regularity conditions have appeared in prior work.Source: §8 · p. 35
  • The discrete method combines piecewise polynomial and Raviart–Thomas spaces with Arnold–Falk–Winther flow elements, yielding well-posedness and a priori estimates.PEERS elements are an alternative for the flow equations; convergence rates are shown for sufficiently smooth solutions.Source: §8 · p. 35
  • Future work includes fully time-dependent analysis, strongly degenerate diffusion, three-dimensional boundary treatment, and reactive-settling applications.The proposed coarser boundary mesh treatment remains to be extended to three dimensions.Source: §8 · p. 35

A.1 Proof of Lemma 4.2

The proof constructs auxiliary fields to control interior and boundary components through divergence and normal-trace conditions, with bounds used to combine the estimates.

A.1 Proof of Lemma 4.2

  • An auxiliary problem defines zψ for the zero-mean component ψ, with normalization ensuring uniqueness and an a priori bound.Setting ςψ := −∇zψ gives div(ςψ) = ψ and zero normal trace.Source: §A.1 · p. 35
  • The boundary component χ is controlled by choosing a unit-norm dual trace η whose pairing with χ bounds its H1/2 norm.An auxiliary problem for zχ then prescribes its normal derivative through η.Source: §A.1 · p. 36
  • The field ςχ := ∇zχ has prescribed normal trace η and a bounded divergence norm, allowing the proof to combine interior and boundary estimates.The text combines the two inequalities after establishing these bounds.Source: §A.1 · p. 36

A.2 Proof of Lemma 4.5

The proof constructs a divergence-conforming field from κ by solving a mixed Poisson problem and uses its estimate to obtain the stated bound.

A.2 Proof of Lemma 4.5

  • The proof tests the estimate with the conjugate-power function κϱ := |κ|^(ρ−2)κ for nonzero κ.Source: §A.2 · p. 36
  • A mixed Poisson problem with Dirichlet data on ΓD and Neumann data on ΓN yields a field eς with divergence κϱ and zero normal trace on ΓN.The construction uses the Friedrichs–Poincaré inequality and an a priori bound for z.Source: §A.2 · p. 36
  • The resulting estimate gives the bound with β2 := (1 + ∥iρ∥/cp)^−1.Source: §A.2 · p. 36

A.3 Proof of Lemma 4.8

The proof builds stress fields with prescribed divergence and symmetry properties to control the components of the test function and derive a bound.

A.3 Proof of Lemma 4.8

  • For w, the proof applies the continuous embedding and Poincaré and Korn inequalities to bound the symmetric gradient of an auxiliary solution.It then constructs a zero-mean-trace stress component τw with divergence ws.Source: §A.3 · p. 37
  • The proof combines the component bounds to obtain a final estimate involving min{C1^−1, C2^−1}.Source: §A.3 · p. 37
  • For the skew-symmetric component eγ, Korn’s inequality bounds an auxiliary symmetric gradient before the proof defines τγ.The resulting stress component has zero divergence.Source: §A.3 · p. 37

A.4 Proof of Lemma 4.11

The proof compares solutions associated with two input pairs through the global bilinear form for (4.1). Lipschitz continuity, regularity, a priori bounds, and the inf-sup condition are used in the estimate.

A.4 Proof of Lemma 4.11

  • The proof begins with solutions to (4.1) corresponding to two input pairs and applies the global bilinear form defined in (4.14).Source: §A.4 · p. 37
  • Lipschitz continuity of D1 and fbk, together with Hölder’s inequality, is used to bound terms in the comparison.Source: §A.4 · p. 38
  • The proof combines the regularity hypothesis, a priori estimates, and the global inf-sup condition to obtain the estimate, with constants independent of the data and input functions.It also uses continuous function-space embeddings and takes a maximum over coefficients multiplying norms.Source: §A.4 · p. 38

A.5 Proof of Lemma 4.13

The proof compares solutions of (4.3) associated with two input pairs using the global bilinear form. Lipschitz continuity, regularity, a priori estimates, and the inf-sup condition yield (4.33).

A.5 Proof of Lemma 4.13

  • For each of two input pairs, the proof takes the corresponding solution of (4.3) and applies the global bilinear form (4.19).Source: §A.5 · p. 38
  • Lipschitz continuity assumptions and Hölder’s inequality with exponents (ρ, r, 2) are used in the comparison.Source: §A.5 · p. 38
  • The proof uses the regularity hypothesis, continuous embeddings, and the a priori estimate before applying the global inf-sup condition to obtain (4.33).The resulting structural constants are absorbed into LS; the constant C3 is independent of the data and input functions.Source: §A.5 · p. 39
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